If you have 3 apple pies and 19 people, how should you slice the pies so that each person gets an equal share? Each person should get 3/19 ≈ 0.1579 pies, but if you make each pie into 6 slices, that’s only 18 slices for 19 people. You have to slice each pie into 6 and 1/3 slices, with each slice being equal except the 1/3 being smaller, and then give the three 1/3 slices to the 19th person.
What if you have 1 pie and the Half-Blood Prince from Harry Potter and the Half-Blood Prince is the only person dining? Then you have 2 pies per person, assuming he’s half a person. In the expression 1/(1/2), move the denominator to the numerator and flip the ex-denominator, making the reciprocal and thereby converting division to multiplication. 1/(1/2) becomes 1*(2/1) which is just 2, because any number without a denominator has a denominator of 1.
Your neighbor lends you $8000 at 3.75% interest compounded annually, with no payments being required for 25 years and the full balance and interest being required to be repaid at that time. What is the payment? $8000*1.0375^25 ≈ $8000*2.5102 = $20,081.34.
What if you want to make a graph of the increasing amount owed on the Cartesian coordinate system where y is the number of years and x is the dollar amount in thousands? Use the equation y = 8*(1.0375^x).
PROBLEM: Your truck gets 15 miles per gallon in the city and 19 mpg on the highway. Your destination is 38 miles away as the crow flies, 42 miles away through the cities, and 49 miles away if you take Interstate-95. However, taking the interstate requires 5 miles of city driving. How many gallons of gas will be used on each route, which one uses the least gas, and how much does each trip cost if gas is $4.15 per gallon?
SOLUTION: First, note that there are only two routes: 42 miles through the cities and 5 miles through the cities PLUS 44 miles on I-95. Then, do this:
42/15 = 2.8 gallons –> 2.8*4.15 = $11.62 in gas
5/15 + 44/19 ≈ 2.65 gallons –> 2.65*4.15 = $10.99 in gas
NOTE that with the variables provided, the longer route is in fact the cheaper route. However, in reality there would be an inordinate number of variables, such as your engine’s efficiency, time of day, traffic patterns, traffic lights, and unforeseen events. For example, traveling an extra 7 miles may necessitate an earlier oil change or some other maintenance. If you get into an accident on the interstate at 80 miles per hour, you might be instantly killed rather than being only wounded if crashing at a much lower speed in a city. Highway driving vs. city driving requires different mental concentration and one may appeal over the other depending on your upbringing and psychological makeup. In the city, you are more likely to be pulled over by policemen and ticketed. The road surface may be smoother on the interstate, which will prevent your tires from wearing down as quickly. If you break down on the interstate, you may be stranded if you don’t have a cell phone. All math problems simplify.
PROBLEM: The 2012 Presidential election is coming up, and observing that Ron Paul (R) and Barack Obama (D) have won the primaries, General Electric is deciding how much to donate to each candidate’s campaign. GE’s budget is $4,000,000, and they estimate Paul has an 8% chance of winning and Obama has a 92% chance of winning. GE estimates donations to Paul have a political worth three times greater than donations to Obama to secure support from the Constitutionalist movement. How much will GE donate to each campaign?
3*0.08 = 0.24
1*0.92 = 0.92
0.24+0.92 = 1.16
4,000,000/1.16 = 3,448,275.862
3,448,275.862 * 0.24 = $827,586.21 to Ron Paul
3,448,275.862 * 0.92 = $3,172,413.79 to Barack Obama
PROBLEM: Steve Jobs is developing the iPad 3 for release November 28, 2012 and must choose between Foxconn’s 64GB isolinear-NAND flash chip and Foxtrot’s 59GB neo-EEPROM flash chip. Foxconn’s chip costs $28.78 and has a five-year failure rate of 2.8%. Foxtrot’s chip costs $26.55 and has a five-year failure rate of 2.1%. Both chips are functionally identical in form factor, read/write speed, power consumption, resiliency, and failure potential, both technologies are equally reliable, both companies use slave labor, and both companies are of equal capacity, reputability, and geographic location.
Apple estimates the market value of an extra 5GB (64GB vs. 59GB) of storage capacity to be $14.50 per unit, and estimates that each five-year failure will have an effective cost of $895.88 on Apple’s image, future sales, and support network. Which chip should Steve Jobs choose?
($28.78 – 14.50) = $14.28 effective cost per 64GB Foxconn chip
$26.55 = $26.55 relative cost 59GB per Foxtrot chip
$895.88 * 0.028 = $25.08 failure cost per 64GB Foxconn chip
$895.88 * 0.021 = $18.81 failure cost per 59GB Foxtrot chip
$14.28 + 25.08 = $39.36 total cost per 64GB Foxconn chip
$26.55 + 18.81 = $45.36 total cost per 59GB Foxconn chip
Jobs should choose the 64GB Foxconn chip, even though it is 0.7% more likely to fail in the first five years, because it’s easier to market a 64GB device than a 59GB device so the 64GB Foxconn chip has an effective cost of $6.00 less than the 59GB Foxtrot chip, given the variables.
PROBLEM: General Motors Company is developing a new type of engine that improves fuel economy by 100%, but has discovered that 0.000097% of the engines blow up when reaching a speed of 88 miles per hour, instantly killing everyone in the vehicle and seriously wounding everyone in a 100 foot radius. GMC is considering including this engine in its new SUV, the ThinkNeighbor Plus, which will get 42 highway miles per gallon instead of the standard 21 highway mpg, will be manufactured in a quantity of 5 million, and will sell for $38,000. GMC estimates only 0.5% of ThinkNeighbor Pluses will ever reach a speed of 88 miles per hour, and estimates the Public Relations costs of each explosion will be $8 million. The U.S. State Department has pledged to blame the explosions on domestic terrorist attacks, but only to a limit of three. Should GMC manufacture the ThinkNeighbor Plus?
NO. Since the question is “should GMC manufacture the ThinkNeighbor Plus?,” it isn’t even a math question because “should” is completely subjective.
More next time.